论文标题
圆和霍奇变性上的ogroupoid结构
Cogroupoid structures on the circle and the Hodge degeneration
论文作者
论文摘要
我们展示了Nonabelian Hodge Theory的Hodge变性，因为在正式模量问题中，过滤环路空间的$ 2 $折叠率$ e_2 $ groupoid。这是一个迭代的groupoid对象，该对象在$ 1 $中恢复了[mrt19]的过滤圆$ s^1_ {fil} $。这在拓扑圆圈上利用了迄今未研究的附加结构，该结构是$ \ \ infty $ - $ - iftty $ - 类别的$ e_2 $ croproupoid对象。我们将这种cogroupoid结构与$ s^1 $上的更常见的“捏合地图”以及lie algebroid $ \ mathbb {t} _ {x} $的todd类相关联；这是在Grothendieck-riemann Roch定理中出现的平滑和正确方案$ x $的不变。特别是，我们将非平凡的TODD类的存在与捏合图的失败是正式的。最后，我们在Hochschild共同学水平上记录了这种结构的一些后果。
We exhibit the Hodge degeneration from nonabelian Hodge theory as a $2$-fold delooping of the filtered loop space $E_2$-groupoid in formal moduli problems. This is an iterated groupoid object which in degree $1$ recovers the filtered circle $S^1_{fil}$ of [MRT19]. This exploits a hitherto unstudied additional piece of structure on the topological circle, that of an $E_2$-cogroupoid object in the $\infty$-category of spaces. We relate this cogroupoid structure with the more commonly studied "pinch map" on $S^1$, as well as the Todd class of the Lie algebroid $\mathbb{T}_{X}$; this is an invariant of a smooth and proper scheme $X$ that arises, for example, in the Grothendieck-Riemann Roch theorem. In particular we relate the existence of non-trivial Todd classes for schemes to the failure of the pinch map to be formal in the sense of rational homotopy theory. Finally we record some consequences of this bit of structure at the level of Hochschild cohomology.
